In conventional quantum mechanics, all unitary evolution takes place within the space-time Hilbert space $\mathcal H_{xt}=L^2(\mathcal M_{xt})$, with time as the sole evolution parameter. The momentum-energy representation $φ(k,E)$ is treated merely as a Fourier re-expression of the same state-kinematically equivalent but dynamically inert. Here we restore the fundamental symmetry between the conjugate pairs $(x,t)$ and $(k,E)$ by extending the quantum theory to an enlarged Hilbert space $\mathcal H_{\text{total}} = \mathcal H_{xt} \oplus \mathcal H_{kE}$, within which the momentum-energy sector $\mathcal H_{kE}=L^2(\mathcal M_{kE})$ carries its own autonomous unitary evolution generated by a self-adjoint operator $\hat{\mathcal T}$. The resulting structure establishes a meta-symmetry: a symmetry between two conjugate dynamical projections of a single global quantum state. It produces a dual-manifold geometry in which each domain is locally complete yet globally open, with divergent limits in one mapping onto extended regions in the other. Remarkably, the dual-manifold symmetry alone reproduces both the uniform dark-energy background and the exponential boundary mapping near black-hole horizons that underlies Hawking radiation. This framework thus opens a quantum-theoretic route to cosmological phenomena that are ordinarily treated within general relativity.
In conventional quantum mechanics, all unitary evolution takes place within the space-time Hilbert space Hxt=L2(Mxt), with time as the sole evolution parameter. The momentum-energy representation φ(k,E) is treated merely as a Fourier re-expression of the same state-kinematically equivalent but dynamically inert. Here we restore the fundamental symmetry between the conjugate pairs (x,t) and (k,E) by extending the quantum theory to an enlarged Hilbert space Htotal=Hxt⊕HkE, within which the momentum-energy sector HkE=L2(MkE) carries its own autonomous unitary evolution generated by a self-adjoint operator T^. The resulting structure establishes a meta-symmetry: a symmetry between two conjugate dynamical projections of a single global quantum state. It produces a dual-manifold geometry in which each domain is locally complete yet globally open, with divergent limits in one mapping onto extended regions in the other. Remarkably, the dual-manifold symmetry alone reproduces both the uniform dark-energy background and the exponential boundary mapping near black-hole horizons that underlies Hawking radiation. This framework thus opens a quantum-theoretic route to cosmological phenomena that are ordinarily treated within general relativity.
The Problem of Time — the apparent incompatibility between quantum mechanics' time parameter t and general relativity's proper time τ — has been treated as a fundamental obstacle to quantum gravity. I argue that this is a level-confusion: both theories define time within their own entity-theoretic frameworks, generating an apparent opposition that disappears when time is relocated to a meta-structural layer. Drawing on the meta-level temporal structure T_SDT := (V_G, ≺) (Zeng 2026), the paper formalizes QM and GR time as two projections of a single Universal Time Formula, parameterized by entity-theoretic tokens. Four token specializations recover the corresponding time concepts. A Bridge Theorem establishes mutual compatibility of all token projections. A bidirectional QM-GR channel is constructed (forward: GR geometry → QM phase; reverse: discrete chain-pair → Lorentzian spacetime), composing into a self-consistent closed loop. Three further results move beyond dissolution toward unification. First, a Madelung-bridge construction derives the Schrödinger equation — Hilbert space, unitarity, Hamiltonian, quantum potential, Born rule — from the determination-graph axioms together with the continuum embedding (CL1), with no independent QM input; the derivation extends to Klein-Gordon. Second, a Planck-scale token μ = t_P / |A_{v'}| is constructed on the discrete 𝒢_D, where |A_{v'}| is the admissibility cardinality; under CL1, this single step-measure recovers QM (|A| ≫ 1) and GR (|A| = 1) as its two continuum limits, with Einstein's equation proved in the classical regime (|A| = 1) via a two-stage argument (Benincasa-Dowker action convergence + standard scalar-tensor variation). The exact quantum-gravitational correction is derived as a scalar-tensor equation with φ = 1/|A|. Third, the discrete determination graph without continuum embedding constitutes a self-contained Planck-scale quantum gravity theory: it has a well-defined path integral, a finite-dimensional S-matrix (unitary in the κ = 0 sector, isometric in the κ > 0 sector; no UV divergences), preserves information across black hole horizons, resolves singularities, derives the Past Hypothesis, and provides structural origins for gauge symmetry, Hawking radiation, spectral dimensionality, and the spin-statistics connection. The Problem of Time is not merely dissolved — it is replaced by a constructive framework in which QM and GR are unified as two limits of a single discrete structure.
In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit i is systematically replaced by the temporal Hilbert transform operator Ht, which satisfies the anti-involution property Ht2=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ and its Hilbert-transformed partner ϕ=Ht[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ, under which the standard U(1) gauge symmetry emerges naturally as a global SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
In the usual formulation of quantum theory, time is a global classical evolution parameter, not a local quantum observable. On the other hand, both canonical quantum gravity (which lacks fundamental time-evolution parameter) and the principle of spacetime covariance (which insists that time should be treated on an equal footing with space) suggest that quantum theory should be slightly reformulated, in a manner that promotes time to a local observable. Such a reformulated quantum theory is unitary in a more general sense than the usual quantum theory. In particular, this promotes the non-unitary Hawking radiation to a unitary phenomenon, which avoids the black-hole information paradox.
The reconstruction of the unitary symmetry \cite{TFD} under non-linear dynamical mapping Hilbert space of action amplitudes CN onto projective Hilbert space CP(N−1) \cite{Le1} has been applied here to the quantum dynamics of elementary vacuum excitations. The "vacuum manifold of virtual action states" is represented here by CP(N−1) whereas its tangent vectors define local dynamical variables (LDV's) describing "matter". The conservation laws of LDV's express self-conservation of the "material particles" during continuous evolution being expressed as the affine parallel transport agrees with Fubuni-Study metric, create the "affine gauge potential" as the solution of the partial differential equations. Such procedure embeds the quantum dynamics into dynamical space-time whose state-dependent coordinates arose due to encoding results of quantum measurement by the qubit spinor whose components subjected to Lorentz transformations of "quantum boosts" and "quantum rotations". Thereby, in the framework of super-relativity, the objective character of the quantum measurement is inherently related to the dynamical space-time structure that replaces the notion of "observer".
Essentially, in this paper we propose a new description of the quantum dynamics by two relativistic propagation wave packets, in the two conjugated spaces, of the coordinates and of the momentum. Compared to the Schrödinger-Dirac equation, which describes a free particle by a wave function continuously expanding in time, considered as the amplitude of a probabilistic distribution of this particle, the new equations describe a free particle as an invariant distribution of matter propagating in the two spaces, as it should be. Matter quantization arises from the equality of the integral of the matter density with the mass describing the dynamics of this density in the phases of the wave packets. In this description, the classical Lagrange and Hamilton equations are obtained as the group velocities of the two wave packets in the coordinate and momentum spaces. When to the relativistic Lagrangian we add terms with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and a scalar potential conjugated to time, we obtain the Lorentz force and the Maxwell equations as characteristics of the quantum dynamics. In this framework, the conventional Schrödinger-Dirac equations of a quantum particle in an electromagnetic field obtain additional terms explicitly depending on velocity, as is expected in the framework of relativistic theory. Such a particle wave function takes the form of a rapidly varying wave, with the frequency corresponding to the rest energy, modulated by the electric potential where this particle is confined, the vector potential of the electromagnetic radiation, and the gravitational potential. In the gravitational field of a black hole, the velocity of a quantum particle is directed to the Schwarzschild boundary from the outside and from the inside, without passing this boundary. When the very strong gravitational field of a large agglomeration of matter has spherical symmetry with a radius smaller than the Schwarzschild radius, the quantum particles in the central region are crushed up to quasi-uniform distributions of matter, exploding with velocities much larger than the light velocity, towards the Schwarzschild boundary, but decelerating in the gravitational field and reaching this boundary only in an infinite time. The matter of a black hole has the tendency to concentrate around the Schwarzschild boundary, with a radius proportional to its mass. In this framework, we believe that our universe is a huge black hole, with the explosion of its central matter as the Big Bang-inflation process, and an increasing radius on account of an absorption of matter from the Total Universe, where our universe has been formed. In a gravitational wave, the time-space interval in the first-order approximation is the product of a gravitational invariant including the amplitude tensor, and a coordinate invariant including the polarization vector. The gravitational invariant describes a rotation of the metric tensor around the direction of the propagation of this wave, with spin 2, as a gravitational quantum called graviton. The coordinate invariant describes a matter rotation with the spins ½ for Fermions, and 1 for Bosons. From the new dynamic equations, for a free particle in the coordinate and momentum spaces, we reobtain the two basic equations of the quantum field theory, but with a change of sign, and an additional term depending on momentum, to the rest mass as the eigenvalue of these equations. However, when these eigenvalues are eliminated, the wave function takes the form of a wave packet of spinors of the same form as in the conventional quantum field theory, with a normalization volume as the integral of the ratio of the energy to the rest energy, over the momentum domain which gives finite dimensions to the quantum particle, as a finite distribution of matter in the coordinate space.